The second overtone of an open pipe A and a closed pipe B have the same frequencies at a given temperature. Both pipes contain air. The ratio of fundamental frequency of A to the fundamental frequency of B is:
(A)`3:5` (B)`5:3` (C)`5:6` (D)`6:5`

Options

A. `3:5`
B. `5:3`
C. `5:6`
D. `6:5`

Answer

To solve the problem, we need to find the ratio of the fundamental frequencies of an open pipe (A) and a closed pipe (B) given that their second overtones are equal.

### Step-by-Step Solution:

1. **Understand the Frequencies of Pipes**:
- The second overtone of an open pipe is the third harmonic, which can be expressed as:
\[ f_{A,3} = 3f_{A,1} \]
where \( f_{A,1} \) is the fundamental frequency of pipe A.

- The second overtone of a closed pipe is the fifth harmonic, expressed as:
\[ f_{B,5} = 5f_{B,1} \]
where \( f_{B,1} \) is the fundamental frequency of pipe B.

2. **Set the Frequencies Equal**:
- According to the problem, the second overtones of both pipes are equal:
\[ f_{A,3} = f_{B,5} \]
- Substituting the expressions for the overtones:
\[ 3f_{A,1} = 5f_{B,1} \]

3. **Rearranging the Equation**:
- Rearranging the equation gives us:
\[ \frac{f_{A,1}}{f_{B,1}} = \frac{5}{3} \]

4. **Conclusion**:
- The ratio of the fundamental frequency of pipe A to that of pipe B is:
\[ \frac{f_{A,1}}{f_{B,1}} = \frac{5}{3} \]
- Therefore, the answer is option (B) \( 5:3 \).